Research projects

Signed distance functions and surface reconstruction

I am interested in researching modern methods of signed distance function processing and surface reconstruction.

Signed distance functions (SDFs) are a very popular surface representation method, and we have recently published two papers (Reach for the Spheres and Reach for the Arcs) that manage to extract much more detail than traditional methods. I am actively researching ways to extract as much information as possible from very little input. I am also interested in researching the theory of SDFs, how to fix invalid SDFs, and AI tasks using SDFs. If you enjoy works like Reach for the Spheres, Reach for the Arcs, or Constructive Solid Geometry on Neural Signed Distance Fields, and are interested in applying information theory and Riemannian geometry to SDFs, you will like working in this area.

The general process of converting one surface representation to another (not just SDFs) is called surface reconstruction. I am actively researching ways to make surface reconstruction more efficient and less lossy. If you like works like Screened Poisson Surface Reconstruction and Iterative Poisson Surface Reconstruction, you will like working in this area.

I have research projects available in both of these research directions, reach out to me if you are interested.

Generative AI via surface deformation

What NanoBanana, ChatGPT and other generative image models have done for image creation, I would like to do for the creation of 3D objects. This is a very active research area, with both commercial and academic approaches that try to generate beautiful 3D objects from simple text prompts. So far, however, all available methods fall short in one way or another, so there is a lot of exciting room for research.

I work in generative AI achieved by deforming an existing object (a triangle mesh, an implicit function, etc.). Our recent work Geometry in Style takes a base shape and then uses a classical geometry processing surface deformation method, driven by a text-based AI, to arrive at an arbitrary target shape. This allows users to design 3D objects purely based on text prompts. I choose to work in this subset of 3D genAI, since it allows me to combine my knowledge of classical geometry processing methods with the cutting edge of AI. I often find that classical graphics insights transfer to the AI world to produce better results.

If you are interested in this research direction, I have research projects available. Reach out to me if you’re interested!

The geometry of computational fabrication

Screens aren’t the only way to output geometry and graphics. We can also manufacture geometry, and thus output our work in the real world. We are all familiar with 3D printers by now: They turn any three-dimensional shape in the computer into a real solid object. I research the geometry of the 3D printing process (it’s much more complicated than you think, and you can’t actually print any shape), but I am also interested in other fabrication methods. Over the history of human civilization, we have developed countless fabrication methods with advantages and disadvantages. My goal is to research the geometric constraints inherent in each fabrication method, and develop design tools that allow people to utilize the potential of any fabrication method to its fullest. Read my works on manufacturing by folding, casting in molds, and sieving. If you find this interesting, and you like working with physical objects, this is the research area for you.

I am also researching reconfigurable shapes. How can we save space and material by using an object for many different tasks? This is a geometry problem, as we must design a shape that can be rearranged, reassembled, or reconfigures to fulfill different constraints in different situations. In a similar vein, I am interested in designing objects specifically for sustainability: How can we make an object that fulfills a certain task, but uses as few resources as possible? This problem has fascinated me for a while, and I have a variety of ideas to solve this problem.

Do you want to design real-world objects using math and algorithms? I have a variety of projects in computational fabrication. Reach out to me to find out more!

Partial differential equations on surfaces

I am interested in numerical analysis, and I research methods for solving Partial differential equations (PDEs) on curved surfaces (i.e., not just flat Euclidean space). I research convergence and stability proofs for existing and novel PDE solution schemes, such as my work on the convergence of the biharmonic equation.

I also research novel and alternative methods for solving PDEs using neural networks instead of classical methods. I research physics-informed neural networks (PINNs) to solve problems previously solved with methods like finite differences or finite elements, and I am working on building a solid mathematical foundation for these novel methods. My recent work on neural networks for fluid simulation allows for the simulation of previously impossible phenomena, and I am currently actively working on improving methods like this and equipping them with mathematical guarantees.

Moreover, I am also interested in computing a variety of mathematical quantities from Riemannian geometry and theoretical physics on curved surfaces. This line of work is less application-oriented, and the fun here is not only in finding discretizations for completely new things, but also in thinking what they might be useful for! In my recent work I have introduced higher-order boundary conditions to geometry processing that, we found, are useful to reduce the bias introduced by boundary geometry to a variety of geometry processing tools (here, here, and here).

If this topic piqued your interest, reach out to me, and I will tell you about the research projects I have available.

Solving classical geometry problems robustly

I am interested in making classical geometry processing problems more robust. While the graphics toolkit is, by now, old and mature, many methods are so fragile that they crash on many inputs. This is especially problematic since, in graphics, users often chain together dozens of operations over a long time, and non-expert users will not provide the methods with perfect inputs. If one operation fails, they might have to restart, losing a lot of valuable work. It also makes it difficult to run geometry tasks unsupervised or to use them for AI.

I have created robust methods (that are guaranteed to work, or have much better success rates than previous methods) for surface parametrization, for the correspondence problem, and for computer animation.

There are many more areas that require robust methods that are just waiting for us to research them! Reach out to me if you want to hear about available projects.